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We first establish that the linearized strains in curvilinear coordinates associated with a given displacement field necessarily satisfy compatibility conditions that constitute the "Saint Venant equations in curvilinear coordinates". We then show that these equations are also sufficient, in the following sense: If a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. In addition, our proof provides an explicit algorithm for recovering such a displacement field from its linearized strains in curvilinear coordinates. This algorithm may be viewed as the linear counterpart of the reconstruction of an immersion from a given flat Riemannian metric.

SAINT VENANT COMPATIBILITY EQUATIONS IN CURVILINEAR COORDINATES